\(q\)-hook formula of Gansner type for a generalized Young diagram. (English. French summary) Zbl 1392.05117

Nakada, Kento

Krattenthaler, Christian (ed.) et al., Proceedings of the 21st annual international conference on formal power series and algebraic combinatorics, FPSAC 2009, Hagenberg, Austria, July 20–24, 2009. Nancy: The Association. Discrete Mathematics & Theoretical Computer Science (DMTCS). Discrete Mathematics and Theoretical Computer Science. Proceedings, 685-696 (2009).

Summary: The purpose of this paper is to present the \(q\)-hook formula of Gansner type for a generalized Young diagram in the sense of D. Peterson and R. A. Proctor. This gives a far-reaching generalization of a hook length formula due to J. S. Frame et al. [Can. J. Math. 6, 316–324 (1954; Zbl 0055.25404)]. Furthurmore, we give a generalization of P. MacMahon’s identity as an application of the \(q\)-hook formula.
For the entire collection see [Zbl 1196.05001].

MSC:

05E10

Combinatorial aspects of representation theory

Keywords:

generalized Young diagrams; trace generating functions; \(q\)-hook formula; Kac-Moody Lie algebra; MacMahon’s identity